[PPT] - Intuitive Beliefs Jawwad Noor Boston University 2 Introduction PowerPoint Presentation

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Intuitive Beliefs

Jawwad Noor Boston University

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Introduction

Economics: rational beliefs described by a prior probability + Bayesian updating
Psychology:

– Beliefs are not monotone let alone additive (conjunction/disjunction fallacy) – Updating is not Bayesian (base-rate neglect, gambler’s fallacy, etc) – Intuitive judgements based on heuristics (Kahneman-Tversky 1974)

Field evidence for non-Bayesian updating

– over/under updating: Stone (EI 2012) – gambler’s fallacy: Suetens et al (JEEA 2016) – Over-reaction: Debondt and Thaler (JF 1985) – Representativeness heuristic: Ahmed and Safdar (MS 2016)

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Introduction

Behavior driven by intuition has economic relevance

– incomplete deliberation due to limited time/cognitive resources – gaps filled by gut feeling

Classic reference on choice based on spontaneous feelings rather than deliberation:

“[A] large proportion of our positive activities...can only be taken as the result of animal spirits—a spontaneous urge to action rather than inaction, and not as the outcome

f a weighted average of quantitative benefits multiplied by quantitative probabilities.”

Keynes (1936)

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Introduction

This paper: formal theory of intuitive beliefs

– Concepually, what is intuition? – How to model mathematically?

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Introduction

This paper: formal theory of intuitive beliefs

– Concepually, what is intuition? – How to model mathematically?

Inspiration from psychology and philosophy: associations
Activation of a mental image activates/inhibits another

– e.g..... – Vocation, ethnicity, religion, etc may bring about the image of a stereotype – Involuntary and costless mental processing

Associations learnable, strength determined by frequency, salience, similarities, etc
Can be strengthened (reinforcement) or weakened (counter-conditioning or decay)
Intuitive beliefs driven by associations triggered by observation of information

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Introduction

This paper: formal theory of intuitive beliefs

– Concepually, what is intuition? – How to model mathematically?

Inspiration from cognitive science: neural networks
Elements of the model (note: not decision-theoretic):

– network of associations, shaped by experience – nodes triggered by an observation generate output at connected nodes – the output of the network is what appears to us as intuition

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Outline

Formal Model
Properties: Evidence, Bayesian intersection, Uniqueness
Axiomatization: Preview
Shaping the network
Conclusion

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Primitives

A state of the world is a description of the relevant uncertainty:
Γ = {  } = finite index set of elements of uncertainty
Ω = {   } = abstract set of possible realizations of element  ∈ Γ
Illustration: Agent is on a boat, lost in the open sea, and is uncertain about

 = health/survival  = what kind of fish is swimming under the water

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Primitives

A state of the world is a description of the relevant uncertainty:
Γ = {  } = finite index set of elements of uncertainty
Ω = {   } = abstract set of possible realizations of element  ∈ Γ
Illustration: Agent is on a boat, lost in the open sea

Γ = { } Ω = {   } Ω = {    }

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Primitives

State space: Ω = Q

∈Γ Ω, generic state  = (  )

Example

Ω = {( ) ( ) }

Large complicated state spaces are not a challenge
Construction and operation of networks is cognitively costless

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Primitives

Σ = {Ω x y z} ⊂ 2Ω = "elementary events" of element 
Example: health/fish in the Yellow Sea...

y ⊂ Ω = {   } y ⊂ Ω = {     }

Event space given by Σ = Q

∈Γ Σ, generic event x = (x x  x)

Identify Yellow Sea with y = (y y) ⊂ Ω × Ω
Product structure is convenient: each elementary event is a node in a network

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Primitives

A belief (·|z) conditional on z ∈ Σ over the space (Ω Σ):

– set function  : Σ → [0 1] – assigns 1 to the full space – 0 to any event with a null elementary event – Satisfies (x|z) = (x ∩ z|z) – Not necessarily additive or monotone (conjunction/disjunction fallacies)

(·|Ω) is the prior, (·|z) for Ω 6=  ∈ Σ is posterior
Primitive: collection of conditional beliefs

p = {(·|z) : z ∈ Σ and (z|Ω)  0}

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Model

Each elementary event is a node of a network
z
{shark}
y
{death}
z
y

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Model

Evaluate ( |y y)
z
{shark}
y
{death}
z
y

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Model

Evaluate ( |y y)
z
{shark}

% %

y −

→ − → − → − → − → • {death}

z
y

(|y) ∈ R+ ∪ {∞} (|y) ∈ R+ ∪ {∞}

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Model

Evaluate ( |y y)
z
{shark}

% &- % &-

y −

→ − → − → − → − → • {death}

z
y

(|y) (|) × (|y) (|y) (|) × (|y)

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Model

Evaluate ( |y y)

↑

z
{shark}

% &- % &-

y −

→ − → − → − → − → • {death} − →

z
y

(|y) (|) × (|y) (|y) (|) × (|y)

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Model

Evaluate ( |y y)

∙

z
{shark}

% ↑ &- % ↑ &-

y −

→ − → ↑ − → − → • {death} ⇒ ↑ % ↑ %

z
y

(|y) (|) × (|y) (|y) (|) × (|y) (|y) (|) × (|y) (|y) (|) × (|y)

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Model

Expression for beliefs: Based on axioms and on tractability

( |y y) =  ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ (|y) (|) × (|y) (|y) (|) × (|y) (|y) (|) × (|y) (|y) (|) × (|y) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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Model

Expression for beliefs: Based on axioms and on tractability

( |y y) = exp−1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ (|y) (|) × (|y) (|y) (|) × (|y) (|y) (|) × (|y) (|y) (|) × (|y) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Inverse of exp so that  ∈ [0 1]

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Model

Expression for beliefs: Based on axioms and on tractability

( |y y) = exp−1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ (|y)−1 (|)−1 × (|y)−1 (|y)−1 (|)−1 × (|y)−1 (|y)−1 (|)−1 × (|y)−1 (|y)−1 (|)−1 × (|y)−1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Inverse of exp so that  ∈ [0 1]
Inverse of signals so that  is increasing in signals

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Model

Expression for beliefs: Based on axioms and on tractability

( |y y) = exp−1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ (|y) (|) × (|y) (|y) (|) × (|y) (|y) (|) × (|y) (|y) (|) × (|y) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Inverse of exp so that  ∈ [0 1]
Inverse of signals so that  is increasing in signals
Define  = 1

,  = 1 

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Model

Expression for beliefs: Based on axioms and on tractability

( |y y) = exp−1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ (|y) + (|) × (|y) +(|y) + (|) × (|y) +(|y) + (|) × (|y) +(|y) + (|) × (|y) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Inverse of exp so that  ∈ [0 1]
Inverse of signals so that  is increasing in signals
Define  = 1

,  = 1  and sum the terms

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Model

Expression for beliefs: Based on axioms and on tractability

( |y y) = exp−1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ (|)(|y) + (|) × (|y) +(|)(|y) + (|) × (|y) +(|)(|y) + (|) × (|y) +(|)(|y) + (|) × (|y) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Inverse of exp so that  ∈ [0 1]
Inverse of signals so that  is increasing in signals
Define  = 1

,  = 1 , sum the terms, let (x|x) = 1

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Model

Expression for beliefs: Based on axioms and on tractability

(x|y) = (x x|y y) = exp−1 ⎡ ⎣X

∈Γ

X

∈Γ

X

∈Γ

(x|x)(x|y) ⎤ ⎦

Inverse of exp so that  ∈ [0 1]
Inverse of signals so that  is increasing in signals
Define  = 1

,  = 1 , sum the terms, let (x|x) = 1

Γ = { }, x = {} x = {}

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Model

Expression for beliefs: Based on axioms and on tractability

(x|y) = (x x|y y) = exp−1 ⎡ ⎣X

∈Γ

X

∈Γ

X

∈Γ

(x|x)(x|y) ⎤ ⎦

Inverse of exp so that  ∈ [0 1]
Inverse of signals so that  is increasing in signals
Define  = 1

,  = 1 , sum the terms, let (x|x) = 1

Γ = { }, x = {} x = {}
To fully unveil the model, suppose the agent sees a shark Evaluate

( |y )

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Model

Information changes: (y y) to (y ) Evaluate ( |y )

∙

z
{shark}

% ↑ &- % ↑ &-

y −

→ − → ↑ − → − → • {death} ⇒ ↑ % ↑ %

z
y
Initial activity in network

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Model

Information changes: (y y) to (y ) Evaluate ( |y )

↑

z
{shark}

& &

y
{death}

⇒ % %

z
y
Now, infinite output at {shark}

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Model

Earlier

( |y y) = exp ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ − ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ (|y) + (|) × (|y) +(|y) + (|) × (|y) +(|y) + (|) × (|y) +(|y) + (|) × (|y) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠

Now set all terms involving shark output to 1

∞ = 0

( |y ) = exp ¡ − £ (|y) + (|y) ¤¢

Drop the element  s.t x = y and sum only over

Γ(x|y) = { ∈ Γ : x & y}

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Model

Earlier

(x|y) = (x x|y y) = exp ⎡ ⎣− X

∈Γ

X

∈Γ

X

∈Γ

(x|x)(x|y) ⎤ ⎦

Model:

(x|y) = (x x|y y) = exp ⎡ ⎣− X

∈Γ

X

∈Γ(x|y)

X

∈Γ(x|y)

(x|x)(x|y) ⎤ ⎦

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Model

Definition. A network is a pair ( ) of mappings that assign a weight

(x|y) (x|y) ∈ R+ ∪ {∞}

to each pair events x y ∈ Σ and elements   ∈ Γ, and satisfies (i)

(Ω|y) = (Ω|y) = ∞

(ii)

(|y) = (|y) = 0

(iii)

(x|x) = 1

Definition. An Intuitive Belief representation for a belief p is a network ( ) such that for any x ∈ Σ and x ⊂ y ∈ Σ+

(x|y) = exp[− X

∈Γ

X

∈Γ(x|y)

X

∈Γ(x|y)

(x|x)(x|y)]

where (x|x) = (x|x)−1 and (x|y) = (x|y)−1

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Properties: Evidence

The model can accommodate Kahneman and Tversky’s findings

– base rate fallacy – conservatism – gamblers fallacy – hot hand fallacy – conjunction fallacy – disjunction fallacy – Contains spirit of Availability heuristic – Unifies Representativeness and Availability heuristics

Unlike the Heuristics and Biases paradigm,

– formal model – empirical connection: frequency data proxies association

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Properties: Bayesian Intuitive Beliefs

Intersection with Bayesian model?
p is (nonadditive) Bayesian if for each  ∈ Σ and  ∈ Σ+,

(x|z) = (x ∩ z|Ω) (z|Ω)

Proposition. Intuitive Beliefs are Bayesian iff they satisfy Statistical Independence:

(x|z) = Y

∈Γ

(x|z)

Proof uses: Let (x x) := P

∈Γ(x|y) (x|x) ≥ 1, then beliefs can be written as

(x|z) = Y

∈Γ

(x|z)(xx)

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Properties: Uniqueness

Direct signals expressed in marginals:

(x|y) = exp " − X

∈Γ

(x|y) #

(x|y) − (x|Ω) is identified by update rule for elementary marginals:

(x|y) = (x|Ω) exp (− [(x|Ω) − (x|y)]) 

Indirect signals expressed in correlation

(xx|y) (x|y)(x|y) = (x|y)(x|x)(x|y)(x|x)

If there exists w s.t. (x|y) = (x|w) then

(xx|y) (x|z)(x|y) (xx|w) (x|z)(x|w) = µ (x|y) (x|w) ¶(x|x)

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Axiomatization: Preview

Axioms for a more general model
Key axioms:

Heuristic Beliefs: For any x ∈ Σ and Γ(x|z) = Γ(x|w)

(x|z) ≥ (x|w) for all  = ⇒ (x|z) ≥ (x|w)

Heuristic Conditioning: For any x ∈ Σ and z w ∈ Σ

(x|z) ≥ (x|w) for all  = ⇒ (x|z) ≥ (x|w)

Joint events hard to evaluate, generated on basis of simpler “elementary marginals”
HB+HC+Separability properties characterize a nested set of models

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Shaping the Network

 is common across y. Additional structure?

(x|z) = exp[− X

∈Γ

X

∈Γ(x|z)

X

∈Γ(x|z)

(x|x)(x|z)]

Hypothesize that network shaped by data/experience
Specifically, “elementary marginals” are matched to objective distribution 

(x|z) = (x|z) = (x∩z|Ω) (z|Ω) = (x∩z|Ω) (z|Ω)

Characterized by restriction that

(x|z)+ X

6=∈Γ

(x|Ω) = (x ∩ z|Ω) − (z|Ω)

Relationship between  and ?

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Related Literature

Heuristics and Biases program of Kahneman-Tversky
Non-Bayesian updating models: IB unifies much of the evidence
Belief formation: Case-based inductive inference (Gilboa and Schmiedler),

Bounded Rationality (Spiegler) – Similarities: (i) Past cases shape the network (ii) frequencies shape beliefs – Differences: (i) Associations can be formed by copies of data as well (ii) Associations do not behave like probabilities

(x|z) 6= (x|y)(y|z)

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Conclusion

Violation of rationality as incomplete deliberation + intuition
Tractable descriptive model, frequency data proxies associations
How does the data shape the network?

– Role of copies, choice, etc. – What will the agent learn asymptotically? – Updating beliefs vs updating network

Application to other domains? Menu-dependent utility, stochastic choice
Rationality and Intuition